1. Pythagorean Theorem

2. Pythagorean Theorem – What is it? For any right triangle, the following relationship will hold:

3. Pythagorean theorem So what do “a”, “b”, and “c” represent? In short the a and b represent the legs of a right triangle and “c” represents the hypotenuse.

4. Pythagorean Theorem E FG Leg which we will label “b” Leg of triangle which we will lablel “a” Hypoteneuse Is always the side across from the right angle Is always the longest side of the right triangle Is always the side labelled “c” ca b

5. Pythagorean Theorem a So if we assign the following value to “a”, “b” and “c” a = 4 b = 3 c = 5 We can see the relationship 3^2 + 4^2 = 5^2 9 + 16 = 25 b c

6. Pythagorean Theorem Graphically, the relationship is illustrated below :

7. Pythagorean Theorem – How Used

8. Pythagorean Triples

9. The triples can “disguised” if they are multiples of the same number 3,4,5 5,12,13 8,15,17 7,24,25 6,8,10 10,24,26 16,30,34 14,48,50 3x,4x,5x 5x,12x,13x 8x,15,17x 7x,24x, 25x

10. Pythagorean Inequalities If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides then the triangle is an acute angle. (If c^2 < a^2 +b^2 then the triangle is acute. ( (Theorem 7.3) If the square of the length of the longest side of a triangle is greater than the sum of the other two sides, then the triangle is an obtuse triangle. (If c^2> a^2 + b^2, the the triangle is obtuse Theorem 7.4) Word of warning…. C should always be the longest leg of the triangle. Pythagorean inequalities can be used to classify triangles as either acute of obtuse (if they are not “right”.